How to Calculate Motor Current With Winding Resistance

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According to Ohm's law, the current (I) through a conducting wire is directly proportional to the applied voltage (V) and the resistance of the wire (R). This relationship doesn't change if the wire is wrapped around a core to form the rotor of an electric motor. In mathematical form, Ohm's law is

V=IR

or, to put current and resistance on different sides of the equal sign:

I=\frac{V}{R}

The wire resistance depends on its diameter, length, conductivity and the ambient temperature. Copper wire is used in most motors, and copper has one of the highest conductivities of any metal.

TL;DR (Too Long; Didn't Read)

Ohm's law tells you that current through a wire – even a long wire wound around a motor solenoid – is equal to the voltage divided by the resistance. You can determine the resistance of a motor coil if you know the wire gauge, the radius of the solenoid and the number of windings.

Wire Resistance

Ohm's law tells you that you can calculate the current through a motor winding if you know the voltage and the resistance of the wire. The voltage is easy to determine. You can attach a voltmeter across the terminals of the power source and measure it. Determining the other variable, wire resistance, isn't as straightforward, because it depends on four variables.

Wire resistance is inversely proportional to wire diameter and conductivity, which means it gets larger as these parameters get smaller. On the other hand, resistance is directly proportional to wire length and temperature – it increases as these parameters increase. To make things even more complicated, conductivity itself changes with temperature. However, if you make your measurements at a particular temperature, such as room temperature, both temperature and conductivity become constants, and you need only consider the length of the wire and its diameter to calculate wire resistance. The resistance (R) becomes equal to a constant (k) multiplied by the ratio of wire length (l) to diameter (d):

R=k\frac{l}{d}

Wire Length and Wire Gauge

You need to know both the length of the wire wrapped around a motor solenoid and the wire's diameter to calculate resistance. However, if you know the wire gauge, you know the diameter, because you can look it up in a table. Some tables help out even further by listing the resistance per standard length for wires of all gauges. For example, the diameter of 16-gauge wire is 1.29 mm or 0.051 inches, and the resistance per 1,000 feet is 4.02 ohms.

At the end of the day, all you really need to measure is the length of the wire, assuming you know the wire gauge. In a motor solenoid, the wire is wrapped multiple times around a core, so to calculate its length, you need two pieces of information: the radius of the core (r) and the number of windings (n). The length of one winding equals the circumference of the core – 2πr – so the total length of the wire is 2πrn. Use this expression to calculate the wire length, and once you know it, you can extrapolate the resistance from a resistance table.

Calculate Current

Knowing the applied voltage and having calculated wire resistance, you have all you need to apply Ohm's law to determine the current flowing through the coil. Because current strength determines the strength of the induced magnetic field of the coil, this information allows you to quantify the power of the motor.

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